Cognitive and Metacognitive Aspects of Mathematical Problem Solving: An Emerging Model
Asmamaw Yimer
University of Wisconsin-Green Bay yimera@ugb.edu
Nerida F. Ellerton
Illinois State University ellerton@ilstu.edu
"Distinguishing between what is cognitive from what is metacognitive has been
problematic (Garofalo & Lester, 1985; Goos & Galbraith, 1996). To help make the study
of metacognition more systematic, frameworks for metacognitive processes associated with
learners’ problem-solving performances across a wide range of domains have been
proposed. For example, Davidson, Deuser and Sternberg (1994) identified four
metacognitive processes that may be applicable in any domain: identifying and defining a
problem; mentally representing the problem; planning how to proceed; evaluating what you
know about your performance. Schoenfeld (1985a) developed a four-stage model which
involved resources, heuristics, control, and belief systems. Garofalo and Lester (1985)
developed a cognitive-metacognitive framework that consisted of four categories:
orientation, organisation, execution, and verification. The cognitive-metacognitive
framework proposed by Artzt and Armour-Thomas (1992) consisted of eight categories:
read, understand, analyse, explore, plan, implement, verify, and watch and listen. Geiger
and Galbraith (1998) developed a script analysis framework that categorised metacognitive
behaviours observed when students solved mathematical problems. Their framework
included engagement, executive behaviours, resources, and beliefs. These models and
frameworks, in fact, all used minor variations of Polya’s (1957) four-stage model —
understand, plan, carry out the plan, and look back." THE MEASUREMENT OF YOUNG PUPILS´ METACOGNITIVE ABILITY IN MATHEMATICS: THE CASE OF
SELF-REPRESENTATION AND SELF-EVALUATION
Areti Panaoura*, George Philippou *edrita@ucy.ac.cy,edphilip@ucy.ac.cy
Department of Education, University of Cyprus
"A demanding view is that “talking about”, as another thought process, should entail more than a simple description of previous thoughts or actions. It could be a metacognitive reflection that involves critical revisiting of the learning processes, in the sense of noting important points of the procedures followed, acknowledging mistakes made on the way, identifying relationships and tracing connections between initial understanding and learning outcomes (Georgiades, 2004)."
"Promoting metacognition begins with building an awareness among learners that metacognition exists, differs from cognition and affects academic success (Schraw, 1998). The first step to attaining insight into ones own mental models is simply getting individuals to become aware of their own processes."
From Secondary Mathematics Syllabuses, Singapore Ministry of Education 3.5 METACOGNITION
Metacognition, or “thinking about thinking”, refers to the awareness of, and
the ability to control one's thinking processes, in particular the selection and
use of problem-solving strategies. It includes monitoring of one's own thinking,
and self-regulation of learning.
The provision of metacognitive experience is necessary to help students
develop their problem solving abilities. The following activities may be used to
develop the metacognitive awareness of students and to enrich their
metacognitive experience:
• Expose students to general problem solving skills, thinking skills and
heuristics, and how these skills can be applied to solve problems.
• Encourage students to think aloud the strategies and methods they use to
solve particular problems.
• Provide students with problems that require planning (before solving) and
evaluation (after solving).
• Encourage students to seek alternative ways of solving the same problem
and to check the appropriateness and reasonableness of the answer.
• Allow students to discuss how to solve a particular problem and to explain
the different methods that they use for solving the problem.
Asmamaw Yimer
University of Wisconsin-Green Bay
yimera@ugb.edu
Nerida F. Ellerton
Illinois State University
ellerton@ilstu.edu
"Distinguishing between what is cognitive from what is metacognitive has been
problematic (Garofalo & Lester, 1985; Goos & Galbraith, 1996). To help make the study
of metacognition more systematic, frameworks for metacognitive processes associated with
learners’ problem-solving performances across a wide range of domains have been
proposed. For example, Davidson, Deuser and Sternberg (1994) identified four
metacognitive processes that may be applicable in any domain: identifying and defining a
problem; mentally representing the problem; planning how to proceed; evaluating what you
know about your performance. Schoenfeld (1985a) developed a four-stage model which
involved resources, heuristics, control, and belief systems. Garofalo and Lester (1985)
developed a cognitive-metacognitive framework that consisted of four categories:
orientation, organisation, execution, and verification. The cognitive-metacognitive
framework proposed by Artzt and Armour-Thomas (1992) consisted of eight categories:
read, understand, analyse, explore, plan, implement, verify, and watch and listen. Geiger
and Galbraith (1998) developed a script analysis framework that categorised metacognitive
behaviours observed when students solved mathematical problems. Their framework
included engagement, executive behaviours, resources, and beliefs. These models and
frameworks, in fact, all used minor variations of Polya’s (1957) four-stage model —
understand, plan, carry out the plan, and look back."
THE MEASUREMENT OF YOUNG PUPILS´ METACOGNITIVE ABILITY IN MATHEMATICS: THE CASE OF
SELF-REPRESENTATION AND SELF-EVALUATION
Areti Panaoura*, George Philippou
*edrita@ucy.ac.cy, edphilip@ucy.ac.cy
Department of Education, University of Cyprus
"A demanding view is that “talking about”, as another thought process, should entail more than a simple description of previous thoughts or actions. It could be a metacognitive reflection that involves critical revisiting of the learning processes, in the sense of noting important points of the procedures followed, acknowledging mistakes made on the way, identifying relationships and tracing connections between initial understanding and learning outcomes (Georgiades, 2004)."
"Promoting metacognition begins with building an awareness among learners that metacognition exists, differs from cognition and affects academic success (Schraw, 1998). The first step to attaining insight into ones own mental models is simply getting individuals to become aware of their own processes."
From Secondary Mathematics Syllabuses, Singapore Ministry of Education
3.5 METACOGNITION
Metacognition, or “thinking about thinking”, refers to the awareness of, and
the ability to control one's thinking processes, in particular the selection and
use of problem-solving strategies. It includes monitoring of one's own thinking,
and self-regulation of learning.
The provision of metacognitive experience is necessary to help students
develop their problem solving abilities. The following activities may be used to
develop the metacognitive awareness of students and to enrich their
metacognitive experience:
• Expose students to general problem solving skills, thinking skills and
heuristics, and how these skills can be applied to solve problems.
• Encourage students to think aloud the strategies and methods they use to
solve particular problems.
• Provide students with problems that require planning (before solving) and
evaluation (after solving).
• Encourage students to seek alternative ways of solving the same problem
and to check the appropriateness and reasonableness of the answer.
• Allow students to discuss how to solve a particular problem and to explain
the different methods that they use for solving the problem.